Metamath Proof Explorer


Theorem wunss

Description: A weak universe is closed under subsets. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wununi.1 ( 𝜑𝑈 ∈ WUni )
wununi.2 ( 𝜑𝐴𝑈 )
wunss.3 ( 𝜑𝐵𝐴 )
Assertion wunss ( 𝜑𝐵𝑈 )

Proof

Step Hyp Ref Expression
1 wununi.1 ( 𝜑𝑈 ∈ WUni )
2 wununi.2 ( 𝜑𝐴𝑈 )
3 wunss.3 ( 𝜑𝐵𝐴 )
4 1 2 wunpw ( 𝜑 → 𝒫 𝐴𝑈 )
5 1 4 wunelss ( 𝜑 → 𝒫 𝐴𝑈 )
6 2 3 sselpwd ( 𝜑𝐵 ∈ 𝒫 𝐴 )
7 5 6 sseldd ( 𝜑𝐵𝑈 )